Orthogonal Matrices - Examples with Solutions

Definition of Orthogonal Matrices

An n × n matrix whose columns form an orthonormal set is called an orthogonal matrix.
As a reminder, a set of vectors is orthonormal if each vector is a unit vector ( length or norm of the vector is equal to n × n and each vector in the set is orthogonal to all other vectors in the set.
These are examples of orthogonal matrices.
Orthogonal Matrices



Properties of Orthogonal Matrices

A list of the most important properties of orthogonal matrices is given below. If Q is an orthogonal matrix, then

  1.    Q-1 = QT ; this is the most important property of orthogonal matrices as the inverse is simply the transpose.
  2.   the rows of Q form an orthonormal set.
  3.    Q-1 is an orthogonal matrix
  4.   Det( Q ) = ~+mn~ 1
  5.   if λ is an eigrnvalue of ( Q ) , then | λ | = 1
  6.   if Q1 and Q2 are n × n orthogonal matrices, then Q1 Q2 is also an orthogonal matrix.



Examples with Solutions

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Example 1
The matrices \( Q_1 = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} \) and \( Q_2 = \begin{bmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1 \end{bmatrix} \) are orthogonal. Verify that the product \( Q_1 Q_2 \) is also orthogonal (Property 6 above)

Solution
We first calculate the product \( Q_1 Q_2 \)
\( Q_1 Q_2 = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1 \end{bmatrix} = \begin{bmatrix} 0&0&-1\\ 0&-1&0\\ 1&0&0 \end{bmatrix} \)

Let \( \textbf v_1 , \textbf v_2 , \textbf v_3 \) be the columns of the matrix \( Q_1 Q_2 \) found above.
\( \textbf {v}_1 = \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix} \) , \( \textbf {v}_2 = \begin{bmatrix} 0\\ -1\\ 0 \end{bmatrix} \) , \( \textbf {v}_3 = \begin{bmatrix} -1\\ 0\\ 0 \end{bmatrix} \)

Let us calculate the length or norm of each column
\( || \textbf {v}_1 || = \sqrt {0^2+0^2+1^2} = 1 \)
\( || \textbf {v}_2 || = \sqrt {0^2+(-1)^2+0^2} = 1 \)
\( || \textbf {v}_3 || = \sqrt {(-1)^2+0^2+0^2} = 1 \)

All three vectors are unit vectors.

Calculate the inner product of all pairs of vectors that can be made from the vectors \( \textbf v_1 , \textbf v_2 , \textbf v_3 \)
\( \textbf {v}_1 \cdot \textbf {v}_2 = 0 \cdot 0 + 0 \cdot (-1) + 1 \cdot 0 = 0 \)
\( \textbf {v}_1 \cdot \textbf {v}_3 = 0 \cdot (-1) + 0 \cdot 0 + 1 \cdot 0 = 0 \)
\( \textbf {v}_2 \cdot \textbf {v}_3 = 0 \cdot (-1) + (-1) \cdot 0 + 0 \cdot 0 = 0 \)
The three vectors form an orthogonal set.
The three columns of the matrix \( Q_1 Q_2 \) are orthogonal and have norm or length equal to 1 and are therefore orthonormal.



Example 2
Use a calculator to find the inverse of the orthogonal matrix \( Q = \begin{bmatrix} 0 & 0 & 1 \\ -1 & 0 & 0 \\ 0 & -1 & 0 \end{bmatrix} \) and verify Property 1 above.

Solution
Use any matrix calculator to find
\( Q^{-1} = \begin{bmatrix} 0 & -1 & 0 \\ 0 & 0 & -1 \\ 1 & 0 & 0 \end{bmatrix} \)

Find the transpose of matrix \( Q \)
\( Q^T = \begin{bmatrix} 0 & -1 & 0 \\ 0 & 0 & -1 \\ 1 & 0 & 0 \end{bmatrix} \)
Hence \( Q^{-1} = Q^T \) , property 1 above.



Example 3
Find the real constants \( a \) and \( b \) in the matrix \( Q = \begin{bmatrix} -1 & 0 & 0 \\ 0 & \dfrac{1}{\sqrt 2} & a \\ 0 & \dfrac{1}{\sqrt 2} & b \end{bmatrix} \) such that \( Q \) is orthogonal.

Solution
Let \( \textbf v_1 , \textbf v_2 , \textbf v_3 \) be the columns of the matrix \( Q \) given above such that
\( \textbf {v}_1 = \begin{bmatrix} -1\\ 0\\ 0 \end{bmatrix} \) , \( \textbf {v}_2 = \begin{bmatrix} 0\\ \dfrac{1}{\sqrt 2}\\ \dfrac{1}{\sqrt 2} \end{bmatrix} \) , \( \textbf {v}_3 = \begin{bmatrix} 0\\ a\\ b \end{bmatrix} \)

Two sets of conditions for matrix \( Q \) to be orthogonal:
1) The norm of each column \( \textbf v_1 , \textbf v_2 , \textbf v_3 \) must equal to 1
\( || \textbf v_1 || = \sqrt {(-1)^2+0^2+0^2} = 1 \)
\( || \textbf v_2 || = \sqrt {0^2+ \left(\dfrac{1}{\sqrt 2} \right)^2+\left(\dfrac{1}{\sqrt 2} \right)^2 } = 1 \)
\( || \textbf v_3 || = \sqrt {0^2+a^2+b^2} = \sqrt {a^2+b^2} = 1 \)         (I)

2) The inner product of any two vectors must be equal to zero (orthogonal vectors)
\( \textbf {v}_1 \cdot \textbf {v}_2 = (-1) \cdot 0 + 0 \cdot \left(\dfrac{1}{\sqrt 2} \right) + 0 \cdot \left(\dfrac{1}{\sqrt 2} \right) = 0 \)
\( \textbf {v}_1 \cdot \textbf {v}_3 = (-1) \cdot 0 + 0 \cdot a + 0 \cdot b = 0 \)
\( \textbf {v}_2 \cdot \textbf {v}_3 = 0 \cdot 0 + \dfrac{1}{\sqrt 2} \cdot a + \dfrac{1}{\sqrt 2} \cdot b = \dfrac{1}{\sqrt 2} \cdot a + \dfrac{1}{\sqrt 2} \cdot b = 0 \)         (II)

For all conditions to be satisfied, we need to solve equations (I) and (II) above
Equation (II) above gives \( a = - b \)

Substitute \( a \) by \( - b \) in equation (I) to obtain
\( \sqrt {2 b^2 } = 1 \)

Solve to obtain
\( b = \pm \dfrac{1}{\sqrt 2} \)

Two solutions to the above question
\( a = - \dfrac{1}{\sqrt 2} \) and \( b = \dfrac{1}{\sqrt 2} \)
\( a = \dfrac{1}{\sqrt 2} \) and \( b = - \dfrac{1}{\sqrt 2} \)



Questions (with solutions given below)

  • Part 1
    Matrix \( Q = \begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \) is orthogonal. Verify the first 5 properties, listed above, for matrix \( Q \).
  • Part 2
    1) Which of the following matrices are orthogonal?
    a) \( A = \begin{bmatrix} \dfrac{1}{\sqrt 2} & \dfrac{1}{\sqrt 2} \\ - \dfrac{1}{\sqrt 2} & \dfrac{1}{\sqrt 2} \end{bmatrix} \) , b) \( B = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{bmatrix} \) , c) \( C = \begin{bmatrix} \dfrac{1}{\sqrt {10}} & - \dfrac{3}{\sqrt {10}} & 0 \\ 0 & 0 & 1 \\ \dfrac{3}{\sqrt {10}} & \dfrac{1}{\sqrt {10}} & 0 \end{bmatrix} \)
  • Part 3
    1) Find all matrices of the form \( A = \begin{bmatrix} p & q\\ \dfrac{1}{\sqrt {3}} & r \end{bmatrix} \) that are orthogonal.
  • Part 4
    Find the inverse of matrix \( A = \begin{bmatrix} 0 & \dfrac{5}{3 \sqrt 5} & \dfrac{2}{3} \\ \dfrac{1}{\sqrt 5} & - \dfrac{4}{3 \sqrt 5} & \dfrac{2}{3} \\ \dfrac{2}{\sqrt 5} & \dfrac{2}{3 \sqrt 5} & - \dfrac{1}{3} \end{bmatrix} \)



Solutions to the Above Questions

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